Perverse sheaves and the topology of algebraic varieties

IAS/Park City Mathematics Series Volume 00, Pages 000 000 S 1079-5634(XX)0000-0 Perverse sheaves and the topology of algebraic varieties Mark Andrea A. de Cataldo Dedicato a Mikki Contents 1 Lecture 1: The decomposition theorem 3 1.1 Deligne s theorem in cohomology 3 1.2 The global invariant cycle theorem 4 1.3 Cohomological decomposition theorem 5 1.4 The local invariant cycle theorem 6 1.5 Deligne s theorem 7 1.6 The decomposition theorem 8 1.7 Exercises for Lecture 1 10 2 Lecture 2: The category of perverse sheaves P(Y) 17 2.1 Three Whys", and a brief history of perverse sheaves 17 2.2 The constructible derived category D(Y) 19 2.3 Definition of perverse sheaves 21 2.4 Artin vanishing and Lefschetz hyperplane theorems 22 2.5 The perverse t-structure 24 2.6 Intersection complexes 25 2.7 Exercises for Lecture 2 26 3 Lecture 3: Semi-small maps 28 3.1 Semi-small maps 29 3.2 The decomposition theorem for semi-small maps 31 3.3 Hilbert schemes of points on surfaces and Heisenberg algebras 31 3.4 The endomorphism algebra End(f Q X ) 33 3.5 Geometric realization of the representations of the Weyl group 34 3.6 Exercises for Lecture 3 35 4 Lecture 4: Symmetries: VD, RHL, IC splits off 37 4.1 Verdier duality and the decomposition theorem 37 4.2 Verdier duality and the decomposition theorem with large fibers 38 4.3 The relative hard Lefschetz theorem 39 Received by the editors July 2, 2015. Partially supported by N.S.F. grant DMS-1301761 and by a grant from the Simons Foundation (#296737 to Mark de Cataldo). 0000 (copyright holder) 1

2 Perverse sheaves and the topology of algebraic varieties 4.4 Application of RHL: Stanley s theorem 40 4.5 Intersection cohomology of the target as a direct summand 41 4.6 Pure Hodge structure on intersection cohomology groups 43 4.7 Exercises for Lecture 4 44 5 Lecture 5: the perverse filtration 46 5.1 The perverse spectral sequence and the perverse filtration 47 5.2 Geometric description of the perverse filtration 48 5.3 Hodge-theoretic consequences 50 5.4 Character variety and Higgs moduli: P = W 50 5.5 Let us conclude with a motivic question 53 5.6 Exercises for Lecture 5 54 Goal of the lectures. The goal of these lectures is to introduce the novice to the use of perverse sheaves in complex algebraic geometry and to what is perhaps the deepest known fact relating the homological invariants of the source and target of a proper map of complex algebraic varieties, namely the decomposition theorem. Notation. A variety is a complex algebraic variety, which we do not assume to be irreducible, nor reduced. We work with cohomology with Q-coefficients as Z-coefficients do not fit well in our story. As we rarely focus on a single cohomological degree, for the most part we consider the total, graded cohomology groups, which we denote by H (X, Q). Bibliographical references. The main reference is the survey [19] and the extensive bibliography contained in it, most of which is not reproduced here. This allowed me to try to minimize the continuous distractions related to the peeling apart of the various versions of the results and of the attributions. The reader may also consult the discussions in [18] that did not make it into the very different final version [19]. Style of the lectures and of the lecture notes. I hope to deliver my lectures in a rather informal style. I plan to introduce some main ideas, followed by what I believe to be a striking application, often with an idea of proof. The lecture notes are not intended to replace in any way the existing literature on the subject, they are a mere amplification of what I can possibly touch upon during the five onehour lectures. As it is usual when meeting a new concept, the theorems and the applications are very important, but I also believe that working with examples, no matter how lowly they may seem, can be truly illuminating and useful in building one s own local and global picture. Because of the time factor, I cannot possibly fit many of these examples in the flow of the lectures. This is why there are plenty of exercises, which are not just about examples, but at times deal headon with actual important theorems. I could have laid-out several more exercises (you can look at my lecture notes [22], or at my little book [9] for more exercises),

Mark Andrea A. de Cataldo 3 but I tried to choose ones that would complement well the lectures; too much of anything is not a good thing anyway. What is missing from these lectures? A lot! Two related topics come to mind: vanishing/nearby cycles and constructions of perverse sheaves; see the survey [19] for a quick introduction to both. To compound this infamy, there is no discussion of the equivariant picture [3]. An afterthought. The 2015 PCMI is now over. Even though I have been away from Mikki, Caterina, Amelie (Amie!) and Dylan for three weeks, my PCMI experience has been wonderful. If you love math, then you should consider participating in future PCMIs. Now, let us get to Lecture 1. 1. Lecture 1: The decomposition theorem Summary of Lecture 1. Deligne s theorem on the degeneration of the Leray spectral sequence for smooth projective maps; this is the 1968 prototype of the 1982 decomposition theorem. Application, via the use of the theory of mixed Hodge structures, to the global invariant cycle theorem, a remarkable topological property enjoyed by families of projective manifolds and compactifications of their total spaces. The main theorem of these lectures, the decomposition theorem, stated in cohomology. Application to a proof of the local invariant cycle theorem, another remarkable topological property concerning degenerations of families of projective manifolds. Deligne s theorem, including semi-simplicity of the direct image sheaves, in the derived category. The decomposition theorem: the direct image complex splits in the derived category into a direct sum of shifted and twisted intersection complexes supported on the target of a proper map. 1.1. Deligne s theorem in cohomology Let us start with a warm-up: the Künneth formula and a question. Let Y, F be varieties. Then (1.1.1) H (Y F, Q) = q 0 H q (Y, Q) H q (F, Q). Note that the restriction map H (Y F, Q) H (F, Q) is surjective. This surjectivity fails in the context of (differentiable) fiber bundles: take the Hopf fibration b : S 3 S 2 (cf. Exercise 1.7.2), for example. It is a remarkable fact that, in the context of algebraic geometry, one has indeed this surjectivity property, and more. Let us start discussing this phenomenon by asking the following Question 1.1.2. Let f : X Y be a family of projective manifolds. What can we say about the restriction maps H (X, Q) H (f 1 (y), Q)? Let X be a projective manifold completing X (i.e. X is open and Zariski-dense in X). What can we say about the restriction maps H (X, Q) H (f 1 (y), Q)? Answer: The answers are given, respectively, by (1.2.1) and by the global invariant cycle Theorem 1.2.2. Both rely on Deligne s Theorem, which we review next.

4 Perverse sheaves and the topology of algebraic varieties The decomposition theorem has an important precursor in Deligne s theorem, which can be viewed as the decomposition theorem in the absence of singularities of the domain, of the target and of the map. We start by stating the cohomological version of his theorem. Theorem 1.1.3. (Blanchard-Deligne 1968 theorem in cohomology [24]) For any smooth projective map 1 f : X Y of algebraic manifolds, there is an isomorphism (1.1.4) H (X, Q) = H q (Y, R q f Q X ), q 0 where R q f Q X denotes the q-th direct image sheaf of the sheaf Q X via the morphism f; see 1.2. More precisely, the Leray spectral sequence (see 1.7) of the map f is E 2 -degenerate. Proof. Exercise 1.7.3 guides you through Deligne s classical trick (the Deligne- Lefschetz criterion) of using the hard Lefschetz theorem on the fibers to force the triviality of the differentials of the Leray spectral sequence. Compare (1.1.1) and (1.1.4): both present cohomological shifts; both express the cohomology of the l.h.s. via cohomology groups on Y; in the former case, we have cohomology with constant coefficients; in the latter, and this is an important distinction, we have cohomology with locally constant coefficients. Deligne s theorem is central in the study of the topology of algebraic varieties. Let us discuss one striking application of this result: the global invariant cycle theorem. 1.2. The global invariant cycle theorem Let f : X Y be a smooth and projective map of algebraic manifolds, let j : X X be an open immersion into a projective manifold and let y Y. What are the images of H (X, Q) and H (X, Q) via the restriction maps into H (f 1 (y), Q)? The answer is the global invariant cycle Theorem 1.2.2 below. The direct image sheaf R q := R q f Q X on Y is the sheaf associated with the pre-sheaf U H q (f 1 (U), Q). In view of Ehresmann s lemma, the proper 2 submersion f is a C fiber bundle. The sheaf R q is then locally constant with stalk R q y = H q (f 1 (y), Q). The fundamental group π 1 (Y, y) acts via linear transformations on R q y: pick a loop γ(t) at y and use a trivialization of the bundle along the loop to move vectors in R q y along R q γ(t), back to Rq y (monodromy action for the locally constant sheaf R q ). Global sections of R q identify with the monodromy invariants (R q y) π 1 R q y. Note, further, that this subspace is defined topologically. The cohomology group 1 Smooth: submersion; projective: factors as X Y P Y (closed embedding, projection). 2 Proper := the pre-image of compact is compact; it is the relative version of compactness.

Mark Andrea A. de Cataldo 5 R q y = H q (f 1 (y), Q) has it own Hodge (p, p )-decomposition (pure Hodge structure of weight q), an algebro-geometric structure. How is (R q y) π 1 R q y placed with respect to the Hodge structure? The E 2 -degeneration Theorem 1.1.3 yields the following immediate, yet, remarkable, consequence: (1.2.1) H q (X, Q) surj (R q y) π 1 R q y = H q (f 1 (y), Q), i.e. the restriction map in cohomology, which automatically factors through the invariants, maps surjectively onto them. The theory of mixed Hodge structures now tells us that the monodromy invariant subspace (R q y) π 1 R q y (a topological gadget) is in fact a Hodge substructure, i.e. it inherits the Hodge (p, p )-decomposition (the algebro-geometric gadget). The same mixed theory implies the highly non-trivial fact (Exercise 1.7.14) that the images of the restriction maps from H (X, Q) and H (X, Q) into H (f 1 (y), Q) coincide. We have reached the following conclusion, proved by Deligne in 1972. Theorem 1.2.2. (Global invariant cycle theorem [26]) Let f : X Y be a smooth and projective map of algebraic manifolds, let j : X X be an open immersion into a projective manifold and let y Y. Then the images of H (X, Q) and H (X, Q) into H (f 1 (y), Q) coincide with the subspace of monodromy invariants. In particular, this latter is a Hodge substructure of the pure Hodge structure H q (f 1 (y), Q). This theorem provides a far-reaching answer to Question 1.1.2. Note that the Hopf examples in Exercise 1.7.2 show that such a nice general answer is not possible outside of the realm of complex algebraic geometry: there are two obstacles, i.e. the non E 2 -degeneration, and the absence of the special kind of global constraints imposed by mixed Hodge structures. 1.3. Cohomological decomposition theorem The decomposition theorem is a generalization of Deligne s Theorem 1.1.3 for smooth proper maps to the case of arbitrary proper maps of algebraic varieties: compare (1.1.4) and (1.3.2). It was first proved by Beilinson-Bernstein-Deligne- Gabber in their monograph [2, Théorème 6.2.5] on perverse sheaves. A possible initial psychological drawback, when compared with Deligne s theorem, is that even if one insists in dealing with maps of projective manifolds, the statement is not about cohomology with locally constant coefficients, but requires the Goresky-MacPherson intersection cohomology groups with twisted coefficients on various subvarieties of the target of the map. However, this is precisely why this theorem is so striking! To get to the point, for now we simply say that we have the intersection cohomology groups IH (S, Q) of an irreducible variety S; they agree with the ordinary cohomology groups when S is nonsingular. The theory is very flexible as it allows for twisted coefficients: given a locally constant sheaf L on a dense open

6 Perverse sheaves and the topology of algebraic varieties subvariety S o S reg S, we get the intersection cohomology groups IH (S, L) of S. We may call such pairs (S, L), enriched varieties (see [39, p.222]; this explains the notation EV below. Theorem 1.3.1. (Cohomological decomposition theorem) Let f : X Y be a proper map of complex algebraic varieties. For every q 0, there is a finite collection EV q of pairs (S, L) with S Y pairwise distinct closed subvarieties of Y, and an isomorphism (1.3.2) IH (X, Q) = IH q (S, L). q 0 (S,L) EV q Note that the same S could appear for distinct q s, hence the notation EV q. Deligne s Theorem 1.1.3 in cohomology is a special case. In particular, we can deduce an appropriate version of the global invariant cycle theorem [2, 6.2.8]. Let us instead focus on its local counterpart. 1.4. The local invariant cycle theorem The decomposition theorem (1.3.2) has a local flavor over the target Y, in both the Zariski and in the classical topology: if we replace Y by an open set U Y, X by f 1 (U), and S by S U, then (1.3.2) remains valid. Let us focus on the classical topology. Let X be nonsingular; this is for the sake of our discussion, for then IH (X, Q) = H (X, Q). Let y Y be a point and let us pick a small Euclidean ball B y Y centered at y, so that (1.3.2) reads: H (f 1 (y), Q) = H (f 1 (B y ), Q) = IH q (S B y, L), q 0,(S,L) EV q where the second equality stems directly from (1.3.2), and the first one can be seen as follows: the constructibility of the direct image complex Rf Q X ensures that the second term can be identified with the stalk (R f Q X ) y, and, in turn, the proper base change theorem ensures that this latter is the first term; see Fact 2.2.1. Let f be surjective. Let f o : X o Y o the restriction of the map f over the open subvariety of Y of regular values for f. Let y o B y be a regular value for f. By looking at Deligne theorem for the map f o it seems reasonable to expect that for every q one of the summands in (1.3.2) should be IH q (Y, L q ), where L q is the locally constant sheaf R q f o Q. This is indeed the case. If follows that for every q 0, IH 0 (B y, L q Y o B y ) is a direct summand of H q (f 1 (y), Q), let us even say that the latter surjects onto the former. Note that we did not assume that y Y o. The intersection cohomology group IH 0 (Y, L q ) is the space of monodromy invariants for the representation π 1 (Y o B y, y o ) GL(H q (f 1 (y o ), Q). Abbreviate the fundamental group notation to π 1,loc. We have reached a very important conclusion: Theorem 1.4.1. (Local invariant cycle theorem, [7] and [2, (6.2.9)] Let f : X Y be a proper surjective map of algebraic varieties with X nonsingular. Let y Y be any

Mark Andrea A. de Cataldo 7 point, let B y be a small Euclidean ball on Y at y, let y o B y be a regular value of f. Then H (f 1 (y), Q) = H (f 1 (B y ), Q) surjects onto the local monodromy invariants H (f 1 (y o ), Q) π 1,loc. 1.5. Deligne s theorem In fact, Deligne proved something stronger than his cohomological theorem (1.1.4), he proved a decomposition theorem for the derived direct image complex under a smooth proper map. Pre-warm-up: cohomological shifts. Given a Z-graded object K = i Z K i, like the total cohomology of a variety, or a complex (of sheaves, for example) on it, or the total cohomology of such a complex, etc., and given an integer a Z, we can shift by the amount a and get a new graded object (with K i+a in degree i) (1.5.1) K[a] := i Z K i+a. If a > 0, then the effect of this operation is to shift K back by a units. Again, if K has non zero entries contained in an interval [m, n], then K[a] has non zero entries contained in [m a, n a]. We have the following basic relation, e.g. for complexes of sheaves H i (K[a]) = H i+a (K). A sheaf F can be viewed as a complex placed in cohomological degree zero; we can then take the F[a] s. We can take a collection of F q s and form q F q [ q], which is a complex with trivial differentials. Then H (Y, q F q [ q]) = q H q (Y, F q ). Warm-up: Künneth for the derived direct image. Let f : X := Y F Y be the projection. Then there is a canonical isomorphism (1.5.2) Rf Q X = q 0 H q (F)[ q] where H q (F) is the constant sheaf on Y with stalk H q (F, Q). The isomorphism takes place in the derived category of the category of sheaves of rational vector spaces on Y; this is where we find the direct image complex Rf Q X, whose cohomology is the cohomology of X: H (Y, Rf Q X ) = H (X, Q). Exercise 1.7.23 asks you to prove (1.5.2). Now to Deligne s 1968 theorem. Theorem 1.5.3. (Deligne s 1968 theorem [24]; semi-simplicity in 1972 [26, 4.2] ) Let f : X Y be a smooth proper map of algebraic varieties. The derived direct image complex has trivial differentials, more precisely, there is an isomorphism in the derived category (1.5.4) Rf Q X = R q f Q [ q]. q 0 Moreover, the locally constant direct image sheaves R q f Q X are semi-simple.

8 Perverse sheaves and the topology of algebraic varieties Equation (1.5.4), which is proved by means of an E 2 -degeneration argument 3 along the lines of the one in Exercise 1.7.3, is the derived version of (1.1.4), which follows by taking cohomology on both sides of (1.5.4). In addition to [24], you may want to consult the first two pages of [30]. The semi-simplicity result is one of the many amazing applications of the theory of weights (Hodge-theoretic, or Frobenius). Terminology and facts about semi-simple locally constant sheaves. To give a locally constant sheaf on Y is the same as giving a representation of the fundamental group of Y (Exercise 1.7.10). By borrowing from the language of representations, we have the notions of simple (no non-trivial locally constant subsheaf; a.k.a. irreducible) and semi-simple (direct sum of simples; a.k.a. completely reducible), indecomposable (no non-trivial direct sum decomposition) locally constant sheaves. Once one has semi-simplicity, one can decompose further. For a semi-simple locally constant sheaf L, we have the canonical isotypic direct sum decomposition (1.5.5) L = χ L χ, where each summand is the span of all mutually isomorphic simple subobjects, and the direct sum ranges over the set of isomorphism classes of irreducible representations of the fundamental group. In particular, in (1.5.4), we have R q f Q X = R q = χ R q χ. What is semi-simplicity good for? Here is the beginning of an answer: look at Exercise 4.7.4, where it is put to good use to give Deligne s proof in [29] of the Hard Lefschetz theorem. In the context of the decomposition theorem, the semi-simplicity of the perverse direct images is an essential ingredient in the proof of the relative hard Lefschetz theorem; see [2] and [16, especially, 5.1 and 6.4]. Keep in mind that the local systems arising naturally in algebraic geometry are typically not semisimple (cf. G. Williamson s Math Overflow post on non semi-simple monodromy in an algebraic family"). 1.6. The decomposition theorem As we have seen, Deligne s theorem in cohomology has a counterpart in the derived category. The cohomological decomposition Theorem 1.3.1 also has a stronger counterpart in the derived category, i.e Theorem 1.6.2. In these lectures, we adopt a version of the decomposition theorem that is more general (and simpler to state!) than the one in [2, 6.2.5] (coefficients of geometric origin) and of [48] (coefficients in polarizable variations of pure Hodge structures). The version we adopt is due essentially to T. Mochizuki [41, 42] (with important contributions of C. Sabbah [47]) and it involves semi-simple coefficients. Mochizuki s [41, 42] deals with projective maps of quasi-projective varieties and 3 People refer to it as the Deligne-Lefschetz criterion.

Mark Andrea A. de Cataldo 9 with C-coefficients; one needs a little bit of tinkering to reach the same conclusions for proper maps of complex varieties with Q-coefficients (to my knowledge, this is not in the literature). Warning: IC vs. IC. We are about to meet the main protagonists of our lectures, the intersection complexes IC S (L) with twisted coefficients; in fact, the actual protagonists are the shifted (see (1.5.1) for the notion of shift): (1.6.1) IC S (L) := IC S (L)[dim S], which are perverse sheaves on S and on any variety Y for which S Y is closed. Note that IC S (L) only has non-trivial cohomology sheaves in the interval [0, dim S 1], the analogous interval for IC S (L) is [ dim S, 1]. Both IC and IC are called intersection complexes. Instead of discussing the pros and cons of either notation, let us move on. Brief on intersection complexes. The intersection cohomology groups of an enriched variety (S, L) are in fact the cohomology groups of S with coefficients in a very special complex of sheaves called the intersection complex of S with coefficients in L and denoted by IC S (L): we have IH (S, L) = H (S, IC S (L)). If S is nonsingular, and L is constant of rank one, then IC S = IC S (Q) = Q S. The decomposition theorem in cohomology (1.3.2) is the shadow in cohomology of a decomposition of the direct image complex Rf IC X in the derived category of sheaves of rational vector spaces on Y. In fact, the decomposition theorem holds in the greater generality of semi-simple coefficients. Theorem 1.6.2. (Decomposition theorem) Let f : X Y be a proper map of complex algebraic varieties. Let IC X (M) be the intersection complex of X with semi-simple twisted coefficients M. For every q 0, there is a finite collection EV q of pairs (S, L) with S pairwise distinct 4 and L semi-simple, and an isomorphism (1.6.3) Rf IC X (M) = q 0 (S,L) EV q IC S (L)[ q]. In particular, by taking cohomology: (1.6.4) IH (X, M) = q,ev q IH q (S, L). We have the isotypic decompositions (1.5.5), which can be plugged into (1.6.4). Remark 1.6.5. The fact that there may be summands associated with S Y should not come as a surprise. It is a natural fact arising from to the singularities (deviation from being smooth) of the map f. One does not need the decomposition theorem to get convinced: the reader can work out the case of the blowing up of the affine plane at the origin; see also Exercise 1.7.20. In general, it is difficult to predict which S will appear in the decomposition theorem; see parts 5 and 7 of Exercise 1.7.21. 4 The same S could appear for distinct q s.

10 Perverse sheaves and the topology of algebraic varieties 1.7. Exercises for Lecture 1 Exercise 1.7.1. (Ehresmann s lemma and local constancy of higher direct images for proper submersions) Let f : X Y be a map of varieties and recall that the q-th direct image sheaf R q := R q f Q X is defined to be the sheafification of the presheaf Y U H q (f 1 (U), Q). If f admits the structure of a C fiber bundle, then the sheaves R q are locally constant, with stalks the cohomology of the fibers. Give examples of maps f : X Y, where the stalks (R q f Q X ) y of the direct image of the constant sheaf differ from the cohomology groups H (f 1 (y), Q) of the corresponding fibers (hint: the maps cannot be proper). If f is a proper smooth map of complex algebraic varieties, then it admits a structure of C fiber bundle (Ehresmann s lemma). Deduce that nonsingular hypersurfaces of fixed degree in complex projective space are all diffeomorphic to each other. Is the same true in real projective space? Why? Quick review of the Leray spectral sequence (see Grothendieck s gem [34], a.k.a. Tohoku ). The Leray spectral sequence for a map f : X Y (and for the sheaf Q X ) is a gadget denoted E pq 2 = H p (Y, R q f Q X ) H p+q (X, Q). There, d 2 r = 0, with r 2 and for which E r+1 = H (E r, d r ). E 2 -degeneration means that d r = 0 for every r 2, so that one has a cohomological decomposition H (X, Q) = q 0 H q (Y, R q f Q X ). Note that with Z coefficients, E 2 -degeneration does not imply the existence of an analogous splitting. are the differentials d r : E pq r E p+r,q r+1 r Exercise 1.7.2. (Maps of Hopf-type) Let a : C 2 \ o P 1 = S 2 be the usual map (x, y) (x : y) from the affine plane with the origin o deleted onto the projective line. It induces maps, b : S 3 S 2 and c : HS := (C 2 \ o)/z P 1 (where 1 Z acts as multiplication by two). These three maps are fiber bundles. Show that there cannot be a cohomological decomposition as in (1.1.4). Deduce that their Leray spectral sequences are not E 2 -degenerate. Observe that the conclusion of the global invariant cycle theorem concerning the surjectivity onto the monodromy invariants fails in all three cases. Exercise 1.7.3. (Proof of the cohomological decomposition (1.1.4) via hard Lefschetz) Let us recall the hard Lefschetz theorem: let X be a projective manifold of dimension d, and let η H 2 (X, Q) be the first Chern class of an ample line bundle on X; then the iterated cup product maps η d q : H q (X, Q) H 2d q (X, Q) are isomorphisms for every q 0. Deduce the primitive Lefschetz decomposition: for every q d, set H q prim := Ker {ηd q+1 : H q H 2d q+2 }; then, for every 0 q d, we have H q = j 0 H q 2j prim, and, for d q 2d, we have H q = η q d ( j 0 H q 2d 2j prim ). Let f : X Y be as in (1.1.4), i.e. smooth and projective and let d := dim X dim Y. Apply the hard Lefschetz theorem to the fibers of the smooth map f and deduce the analogue of the primitive Lefschetz decomposition for the direct image sheaves R q := Rf Q X. Argue that in order to deduce (1.1.4) it is enough to show the differentials d r of the spectral sequence

Mark Andrea A. de Cataldo 11 vanish on H p (Y, R q prim ) for every q d. Use the following commutative diagram, with some entries left blank on purpose for you to fill-in, to deduce that indeed we have the desired vanishing: H? (Y, R q prim ) d? H? (Y,?) η? H? (Y,?) η? d? H? (Y,?). (Hint: the right power of η kills a primitive class in degree q, but is injective in degree q 1.) Remark: the refined decomposition (1.5.4) is proved in a similar way by replacing the spectral sequence above with the analogous one for Hom(R q [ q], Rf Q X ): first you prove it is E 2 -degenerate; then you lift the identity R q R q to a map in Hom(R q [ q], Rf Q X ) inducing the identity on R q ; see [30]. Heuristics for E 2 -degeneration and for semi-simplicity of the R q f Q X via weights. (What follows should be considered as a very informal fireside chat.) It seems that Deligne guessed at E 2 -degeneration by looking at the same situation over the algebraic closure of a finite field by considerations ( the yoga of weights [25, 28]) of the size (weight) of the eigenvalues of the action of Frobenius on the entries E pq r : they should have weight something analogous to exp (p + q) (we are using the exponential function as an analogy only, one needs to say more, but we shan t) so the Frobenius-compatible differentials must be zero. There are similar heuristic considerations for Deligne s theorem to the effect that the R q are semisimple: if 0 M R q N 0 is a short exact sequence, then Frobenius acts on Ext 1 (N, M) with weight exp (1); take M R q to be the maximal semi-simple subobject; then the corresponding extension is invariant under Frobenius and has weight zero (1 = exp (0)); it follows that the extension splits and the biggest semi-simple in R q splits off: R q = M N; if the resulting quotient N were nontrivial, then it would contain a non-trivial simple that then, by the splitting, would enlarge the biggest semi-simple M in R q ; contradiction. This kind of heuristic is now firmly based in deep theorems by Deligne and others [2, 29] for varieties over finite fields and their algebraic closure, and by M. Saito [48] in the context of mixed Hodge modules over complex algebraic varieties. Exercise 1.7.4. (Rank one locally constant sheaves) Take [0, 1] Q and identify the two ends, {0} Q and {1} Q, by multiplication by 1 Q. Interpret this as a rank one locally constant sheaf on S 1 that is not constant. Do the same, but multiply by 2. Do the same, but first replace Q with Q and multiply by a root of unity. Show that the tensor product operation (L, M) L M induces the structure of an abelian group on the set of isomorphisms classes of rank one locally constant sheaves on S 1. Determine the torsion elements of this group when you replace Q with Q. Show that if we replace S 1 with any connected

12 Perverse sheaves and the topology of algebraic varieties variety and Q with C, then we obtain the structure of a complex Lie group (a.k.a. the character variety for rank one complex representations; one can define it for arbitrary rank, but needs geometric invariant theory to do so). Exercise 1.7.5. (Locally constant sheaves and representations of the fundamental group) A locally constant sheaf (a.k.a. a local system) L on Y gives rise to a representation ρ L : π 1 (Y, y) GL(L y ): pick a loop γ(t) at y and use local trivializations of L along the loop to move vectors in L y along L γ(t), back to L y. Exercise 1.7.6. (Representations of the fundamental group and locally constant sheaves) Given a representation ρ : π 1 (Y, y) GL(V) into a finite dimensional vector space, consider a universal cover (Ỹ, ỹ) (Y, y), build the quotient space (V Ỹ)/π 1(Y, y), take the natural map (projection) to Y and take the sheaf of its local sections. Show that this is a locally constant sheaf whose associated representation is ρ. Exercise 1.7.7. (Zeroth cohomology of a local system) Let X be a connected space. Let L be a local system on X, and write M for the associated π 1 (X)-representation. Show that H 0 (X; L) = M π 1(X), where the right hand side is the fixed part of M under the π 1 (X)-action. Exercise 1.7.8. (Cohomology of local systems on a circle) Fix an orientation of S 1 and the generator T π 1 (S 1 ) that comes with it. Let L be a local system on S 1 with associated monodromy representation M. Show that H 0 (S 1, L) = ker((t id): M M), H 1 (S 1, L) = coker((t id): M M), and that H >1 (S 1, L) = 0. (Hint: one way to proceed is to use Cech cohomology. Alternatively, embed S 1 as the boundary of a disk and use relative cohomology (or dualize and use compactly supported cohomology, where the orientation is easier to get a handle on)). Exercise 1.7.9. (Fiber bundles over a circle: the Wang sequence) This is an extension of the previous exercise. Let f: E S 1 be a locally trivial fibration with fibre F and monodromy isomorphism T : F F. Show that the Leray spectral sequence gives rise to a short exact sequence 0 H 1 (S 1, R q 1 f Q) H q (E, Q) H 0 (S 1, R q f Q) 0. Use the previous exercise to put these together into a long exact sequence... H q (E, Q) H q (F, Q) H q (F, Q) H q+1 (E)... where the middle map H q (F; Q) H q (F; Q) is given by T id. Make a connection with the theory of nearby cycles (which are not discussed in these notes). Exercise 1.7.10. (The abelian category Loc(Y)) Show that the abelian category Loc (Y) of locally constant sheaves of finite rank on Y is equivalent to the abelian

Mark Andrea A. de Cataldo 13 category of finite dimensional π 1 (Y, y)-representations. Show that both categories are noetherian (acc ok!), artinian (dcc ok!) and have a duality anti-selfequivalence. Exercise 1.7.11 below is in striking contrast with the category Loc, but also with the one of perverse sheaves, which admits, by its very definition, the antiself-equivalence given by Verdier duality. Exercise 1.7.11. (The abelian category Sh c (Y) is not artinian.) Show that in the presence of such an anti-self-equivalence, noetherian is equivalent to artinian. Observe that the category Sh c (Y) whose objects are the constructible sheaves (i.e. there is a finite partition of Y = Y i into locally closed subvarieties to which the sheaf restricts to a locally constant one (always assumed to be of finite rank!) is abelian and noetherian, but it is not artinian. Deduce that Sh c (Y) does not admit an anti-self-equivalence. Give an explicit example of the failure of dcc in Sh c (Y). Prove that Sh c (Y) is artinian if and only if dim Y = 0. Exercise 1.7.12. (Cyclic coverings) Show that the direct image sheaf sheaf R 0 f Q for the map S 1 S 1, t t n is a semi-simple locally constant sheaf of rank n; find its simple summands (one of them is the constant sheaf Q S 1 and the resulting splitting is given by the trace map). Do the same for R 0 f Q. Exercise 1.7.13. (Indecomposable non simple) The rank two locally constant sheaf on S 1 given by the non-trivial unipotent 2 2 Jordan block is indecomposable and is neither simple nor semi-simple. Make a connection between this locally constant sheaf and the Picard-Lefschetz formula for the degeneration of a curve of genus one to a nodal curve. Amusing monodromy dichotomy. There is an important and amusing dichotomy concerning local systems in algebraic geometry (which we state informally): the global local systems arising in complex algebraic geometry via the decomposition theorem are semi-simple (i.e. completely reducible; related to the Zariski closure of the image of the fundamental group in the general linear group being reductive); the restriction of these local systems to small punctured disks with centers at infinity (degenerations), are quasi-unipotent, i.e. unipotent after taking a finite cyclic covering if necessary. This local quasi-unipotency is in some sense the opposite of the global complete reducibility. Quick review of Deligne s 1972 and 1974 theory of mixed Hodge structures [25 27]; see also [31]. Deligne discovered the existence of a remarkable structure, a mixed Hodge structure, on the singular cohomology of a complex algebraic variety X (not necessarily smooth, complete or irreducible): there is an increasing filtration W k H (X, Q) and a decreasing filtration F p H (X, C) (with conjugate filtration denoted by F) such that the graded quotients Gr W k H (X, C) = p+q=k H pq k, where the splitting is induced by the (conjugate and opposite) filtrations F, F; i.e. (W, F, F) induce pure Hodge structure of weight k on Gr W k. This structure is

14 Perverse sheaves and the topology of algebraic varieties canonical and functorial for maps of complex algebraic varieties. Important: one has that a map of mixed Hodge structures f : A B is automatically strict, i.e. if f(a) W k B, then there is a W k A with f(a ) = f(a). Kernels, images and cokernels of pull-back maps in cohomology inherit such a structure. If X is a projective manifold, we get the known Hodge (p, q)-decomposition: H i (X, C) = p+q=i H pq (X). It is important to take note that for each fixed i we have H i (X, C) = k Gr W k Hi (X, C) = k p+q=k H pq k (X) which may admit several non zero k summands for k i. In this case, we say that the mixed Hodge structure is mixed. This happens for the projective nodal cubic: H 1 = H 0,0 0, and for the punctured affine line H 1 = H 1,1 2. Here are some inequalities for the weight filtration: Gr W k Hd = 0 for k / [0, 2d]; if X is complete, then Gr W k>d Hd = 0; if X is nonsingular, then Gr W k<d Hd = 0 and W d H d is the image of the restriction map from any nonsingular completion (open immersion in proper nonsingular); if X Y is surjective and X is complete nonsingular, then the kernel of the pullback to H d (X) is W d 1 H d (Y). Exercise 1.7.14. (Amazing weights) Let Z U X be a closed immersion with Z complete followed by an open dense immersion into a complete nonsingular variety. Use some of the weight inequalities listed above, together with strictness, to show that the images of H (X, Q) and H (U, Q) into H (Z, Q) coincide. Build a counterexample in complex geometry (Hopf!). Build a counterexample in real algebraic geometry (circle, bi-punctured sphere, sphere). The reader is invited to produce an explicit example of a projective normal surface having mixed singular cohomology. Morally speaking, as soon as you leave the world of projective manifolds and dive into the one of projective varieties, mixedness is the norm. For an explicit example of a proper map with no cohomological decomposition analogous to (1.1.4), see Exercise 1.7.18. We can produce many by pure-thought using Deligne s theory of mixed Hodge structures. Here is how. Exercise 1.7.15. (In general, there is no decomposition Rf Q X = R q f Q X [ q]) Pick a normal projective variety Y whose singular cohomology is a non pure mixed Hodge structure. Resolve the singularities f : X Y. Use Zariski s main theorem to show that R 0 f Q X = Q Y. Show that, in view of the the mixed-notpure assumption, the map of mixed Hodge structures f is not injective. Deduce that Q Y is not a direct summand of Rf Q X and that, in particular, there is no decomposition Rf Q X = R q f Q X [ q] in this case. (In some sense, the absence of such a decomposition is the norm for proper maps of varieties.) Exercise 1.7.16. (The affine cone Y over a projective manifold V) Let V d P be an embedded projective manifold of dimension d and let Y d+1 A be its affine cone with vertex o. Let j : U := Y \ {o} Y be the open embedding. Show that U is the C -bundle over V of the dual to the hyperplane line bundle for the given embedding V P. Determine H (U, Q). Answer: for every for 0 q

Mark Andrea A. de Cataldo 15 d, H q (U) = H q prim (V) and H1+d+q (U) = H d q prim (V). Show that R0 j Q U = Q Y and that, for q > 0, R q j Q U is skyscraper at o with stalk H q (U, Q). Compute H q c (Y, Q). Give a necessary and sufficient condition on the cohomology of V that ensures that Y satisfies Poincaré duality H q (Y, Q) = Hc 2d+2 q (Y, Q). Observe that if V is a curve this condition boils down to it having genus zero. Remark: once you know about a bit about Verdier duality, this exercise tells you that the complex Q Y [dim Y] is Verdier self dual if and only if V meets the condition you have identified above; in particular, it does not if V is a curve of positive genus. Fact 1.7.17. (IC Y, Y a cone over a projective manifold V) Let things be as in Exercise 1.7.16. By adopting the definition of the intersection complex as an iterated push-forward followed by truncations, as originally given by Goresky- MacPherson, the intersection complex of Y is defined to be IC Y := τ d Rj Q U, where we are truncating the image direct complex Rj Q U in the following way: keep the same entries up to degree d 1, replace the d-th entry by the kernel of the differential exiting it and setting the remaining entries to be zero; the resulting cohomology sheaves are the same as the ones for Rj Q U up to degree d included, and they are zero afterwards. More precisely, the cohomology sheaves of this complex are as follows: H q = 0 for q / [0, d], H 0 = Q Y, and for 1 q d, H i is skyscaper at o with stalk H q prim (V, Q). Here is a justification for this definition: while Q[dim Y] usually fails to be Verdier self-dual, one can verify directly that the intersection complex IC Y := IC Y [dim Y] is Verdier self-dual. In general, if we were to truncate at any other spot, then we would not get this self-duality behavior (unless we truncate at minus 1 and get zero). Note that the knowledge of the cohomology sheaves of a complex, e.g. IC Y, is important information, but it does not characterize the complex up to isomorphism. Exercise 1.7.18. (Example of no decomposition Rf Q X = R q f Q X [ q]) Let things be as in Exercise 1.7.16 and assume that V is a curve of positive genus, so that Y is a surface. Let f : X Y be the resolution obtained by blowing up the vertex o Y. Use the failure of the self-duality of Q Y [2] to deduce that Q Y is not a direct summand of Rf Q X. Deduce that Rf Q X R q f Q X [ q]. (The reader is invited to check [17, 3.1] out: it contains an explicit computation dealing with this example showing that as you try split Q Y off Rf Q X, you meet an obstruction; instead, you end up splitting IC Y off Rf Q X, provided you have defined IC Y as the truncated push-forward as above.) Fact 1.7.19. (Intersection complexes on curves) Let Y o be a nonsingular curve and L be a locally constant sheaf on it. Let j : Y o Y be an open immersion into another curve (e.g. a compactification). Then IC Y (L) = j L[1] (definition of IC via push-forward/truncation). Note that if y Y is a nonsingular point, then the stalk (j L) y is given by the local monodromy invariants of L around a small loop about y. The complex Rj L[1] may fail to be Verdier self-dual, whereas its truncation τ 1 Rj L[1] = j L[1] = IC Y (L) is Verdier self-dual. Note that we have

16 Perverse sheaves and the topology of algebraic varieties a factorization Rj! L[1] IC Y (L) Rj L[1]. This is not an accident; see the end of 2.6. Exercise 1.7.20. (Blow-ups) Compute the direct image sheaves R q f Q for the blowup of C m C n (start with m = 0; observe that there is a product decomposition of the situation that allows you to reduce to the case m = 0). Same question for the composition of the blow up of C 1 C 3, followed by the blowing up of a positive dimensional fiber of the first blow up. Observe that in all cases, one gets an the decomposition Rf Q = R q f Q[ q]. Guess the shape of the decomposition theorem in both cases. Exercise 1.7.21. (Examples of the decomposition theorem) Guess the exact form of the cohomological and derived decomposition theorem in the following cases: 1) the normalization of a cubic curve with a node and of a cubic curve with a cusp; 2) the blowing up of a smooth subvariety of an algebraic manifold; 3) compositions of various iterations of blowing ups of nonsingular varieties along smooth centers; 4) a projection F Y Y; 5) the blowing up of the vertex of the affine cone over the nonsingular quadric in P 3 ; 6) same but for the projective cone; 7) blow up the same affine and projective cones but along a plane through the vertex of the cone; 8) the blowing up of the vertex of the affine/projective cone over an embedded projective manifold. Exercise 1.7.22. (decomposition theorem for Lefschetz pencils) Guess the shape of the decomposition theorem for a Lefschetz pencil f : X P 1 on a nonsingular projective surface X. Work out explicitly the invariant cycle theorems in this case. Do the same for a nonsingular projective manifold. When do we get skyscraper contributions? Exercise 1.7.23. (Künneth for the derived image complex) One needs a little bit of working experience with the derived category to carry out what follows below. But try anyway. Let f : X := Y F Y. A class a q H q (X, Q) is the same thing as a map in the derived category a q : Q X Q X [q]. First pushing forward via Rf, then observing that Rf f Q Y = Rf Q X, and, finally, pre-composing with the adjunction map Q Y Rf Q X, yields a map a q : Q Y Rf Q X [q]. Take a q to be of the form pr F α q. Obtain a map α q : H q (F) Rf Q X [q]. Next, shift this map to get α q : H q (F)[ q] Rf Q X. Show that the map induces the identity on the q-th direct image sheaf and zero on the other direct image sheaves. Deduce that q α q : q H q (F)[ q] Rf Q X is an isomorphism in the derived category inducing the identity on the cohomology sheaves. Observe that you did not make any choice in what above, i.e. the resulting isomorphism is canonical, whereas in Deligne s theorem 1.5.3, one does not obtain a canonical isomorphism. Exercise 1.7.24. (Deligne s theorem as a special case of the decomposition theorem) Keeping in mind that if S o = S, then IC S (L) = L, recover the Deligne theorem from the decomposition theorem.

Mark Andrea A. de Cataldo 17 2. Lecture 2: The category of perverse sheaves P(Y) Summary of Lecture 2. The constructible derived category. Definition of perverse sheaves. Artin vanishing and its relation to a proof of the Lefschetz hyperplane theorem for perverse sheaves. The perverse t-structure (really, only the perverse cohomology functors!). Beilinson s and Nori s equivalence theorems. Several equivalent definitions of intersection complexes. 2.1. Three Whys", and a brief history of perverse sheaves Why intersection cohomology? Let us look at (1.3.2) for X and Y nonsingular: H (X, Q) = q,evq IH q (S, L), i.e. the l.h.s. is ordinary cohomology, but the r.h.s. is not any kind of ordinary cohomology on Y: we need intersection cohomology to state the decomposition theorem, even when X and Y are nonsingular. The intersection cohomology groups of a projective variety enjoy a battery of wonderful properties (Poincaré-Hodge- Lefschetz package). In some sense, intersection cohomology nicely replaces singular cohomology on singular varieties, but with a funny twist: singular cohomology is functorial, but has no Poincaré duality; intersection cohomology has Poincaré duality, but is not functorial! Why the constructible derived category? The cohomological Deligne theorem (1.1.4) for smooth projective maps is a purely cohomological statement and it can be proved via purely cohomological methods (hard Lefschetz + Leray spectral sequence). The cohomological decomposition theorem (1.3.2) is also a cohomological statement. However, there is no known proof of this statement that does not make use of the formalism of the middle perversity t-structure present in the constructible derived category: one proves the derived version (1.6.3) and then deduces the cohomological one (1.3.2) by taking cohomology. Actually, the definition of perverse sheaves does not make sense if we take the whole derived category, we need to take complexes with cohomology sheaves supported at closed subvarieties (not just classically closed subsets). We thus restrict to an agreeable, yet flexible, class of complexes: the constructible complexes. Why perverse sheaves? Intersection complexes, i.e. the objects appearing on both sides of the decomposition theorem (1.6.3) are very special perverse sheaves. In fact, in a precise way, they form the building blocks of the category of perverse sheaves: every perverse sheaf is an iterated extension of a collection of intersection complexes. Perverse sheaves satisfy their own set of beautiful properties: Artin vanishing theorem, Lefschetz hyperplane theorem, stability via duality, stability via vanishing and nearby cycle functors. As mentioned above, the known proofs of the decomposition theorem use the machinery of perverse sheaves. A brief history of perverse sheaves. Intersection complexes were invented by Goresky-MacPherson as a tool to systematize, strengthen and widen the scope of their own intersection cohomology

18 Perverse sheaves and the topology of algebraic varieties theory. For example, their original geometric proof of Poincaré duality can be replaced by the self-duality property of the intersection complex. See also S. Kleiman s very entertaining survey [38]. The conditions leading to the definition of perverse sheaves appeared first in connection with the Riemann-Hilbert correspondence established by Kashiwara and by Mekbouth: their result is an equivalence of categories between the constructible derived category (which we have been procrastinating to define) and the derived category of regular holonomic D-modules (which we shall not define); the standard t-structures, given by the standard truncations met in Exercise 1.7.17, of these two categories do not correspond to each other under the Riemann-Hilbert equivalence; the conditions leading to the conditions of support defining of perverse sheaves are the (non-trivial) translation in the constructible derived category of the conditions on the D-module side stating that a complex of D-modules has trivial cohomology D-modules in positive degree. It is a seemingly unrelated, yet remarkable and beautiful fact, that the conditions of support so obtained are precisely what makes the Artin vanishing Theorem 2.4.1 work on an affine variety. As mentioned above, Gelfand and MacPherson conjectured the decomposition theorem for Rf IC X. Meanwhile, Deligne had developed a theory of pure complexes for varieties defined over finite fields and established the invariance of purity under push-forward by proper maps. Gabber proved that the intersection complex of a pure local system, in that context, is pure. The four authors of [2] introduced and developed systematically the basis for the theory of t-structures, especially with respect to the middle perversity. They then proved that the notions of purity and perverse t-structure are compatible: a pure complex splits over the algebraic closure of the finite field as prescribed by the r.h.s. of (1.6.3). The decomposition theorem over the algebraic closure of a finite field follows when considering the purity result for the proper direct image mentioned above. The whole of Ch. 6 in [2], aptly named De F à C, is devoted to explaining how these kind of results over the algebraic closure of a finite field yield results over the field of complex numbers. This established the original proof of the decomposition theorem over the complex numbers for semi-simple complexes of geometric origin (see [2, 6.2.4, 6.2.5]), such as IC X. M. Saito has developed in [48] the theory of mixed Hodge modules which yields the desired decomposition theorem when M underlies a variation of polarizable pure Hodge structures. M. A. de Cataldo and L. Migliorini have given a proof based on classical Hodge theory of the decomposition theorem when M is constant [16]. Finally, the decomposition theorem stated in (1.6.3) is the most general statement currently available over the complex numbers and is due to work of C. Sabbah [47] and T. Mochizuki [41, 42] (where this is done in the essential case of projective maps of quasi projective manifolds; it is possible to extend it to proper

Mark Andrea A. de Cataldo 19 maps of algebraic varieties). The methods (tame harmonic bundles, D-modules) are quite different from the ones discussed in these lectures. 2.2. The constructible derived category D(Y) The decomposition theorem isomorphisms (1.6.3) take place in the constructible derived category D(Y). It is probably a good time to try and give an idea what this category is. Constructible sheaf. A sheaf F on Y is constructible if there is a finite disjoint union decomposition Y = a S a into locally closed subvarieties such that the restriction F Sa are locally constant sheaves of finite rank. This is a good time to look at Exercise 2.7.1. Constructible complex. A complex C of sheaves of rational vector spaces on Y is said to be constructible if it is bounded (all but finitely many of its cohomology sheaves are zero) and its cohomology sheaves are constructible sheaves. See the most-important Fact 2.2.1. Constructible derived category. The definition of D(Y) is kind of a mouthful: it is the full subcategory of the derived category D(Sh(Y, Q)) of the category of sheaves of rational vector spaces whose objects are the constructible complexes. It usually takes time to absorb these notions and to absorb the apparatus it gives rise to. We take a different approach and we try to isolate some of the aspects of the theory that are more relevant to the decomposition theorem. We do not dwell on technical details. Cohomology. Of course, the first functors to consider are cohomology and cohomology with compact supports H i (Y, ), H i c(y, ) : D(Y) D(point). They can be seen as special cases of derived direct images. Derived direct images Rf, Rf!. We can define derived direct image maps Rf, Rf! : D(X) D(Y), for every map f : X Y. The first thing to know is that H (X, C) = H (Y, Rf C) and that H c(x, C) = H c(y, Rf! C), so that we may view them as generalizing cohomology. Pull-backs. The pull-back functor f is probably the most intuitive one. The extraordinary pull-back functor f! is tricky and we will not dwell on it. It is the right adjoint to Rf! ; for open immersions, f! = f ; for closed immersions it is the derived version of the sheaf of sections supported on the closed subvariety; for smooth maps of relative dimension d, f! = f [2d]. A down-to-earth reference for f! and duality I like is [35] (good also, among other things, as an introduction to Borel-Moore homology). I also like [32]. There is also the seemingly inescapable, and nearly encyclopedic [37]. Fact 2.2.1. A good reference is [4]. Given C D(Y), and y Y there is a system of standard neighborhoods U y (ɛ) (think of 0 < ɛ 1 as the radius of an Euclidean ball; of course, our U y (ɛ) are singular, if Y is singular at y) such that H (U y (ɛ), C) and H c(u y (ɛ), C) are constant (make the meaning of constant precise) for 0 < ɛ 1. The U y (ɛ) are cofinal in the system of neighborhoods of